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# geometry

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In pentagon ABCDE ,  BC = CD = DE = 2 units, angle E is a right angle and angle B = angle C = angle D = 60 . The length of segment  can be expressed in simplest radical form as AE = a + b*sqrt(c) units. What is the value of a + b + c?

Sep 5, 2021

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iirc, this is very similar to a previous mathcounts nationals problem. here is the solution to that from my document

CFD is a 90, 45, 45 triangle, so CF and DF is $\sqrt2$
$BE = \sqrt(2(2+\sqrt(2))^2) = \sqrt{2} \cdot (2 + \sqrt{2})= 2\sqrt{2} + 2$
$AE^2 = AB^2 + BE^2 = 2 \times (2\sqrt{2} + 2)^2$
$AE = \sqrt{2} \times (2\sqrt{2} + 2) = 4 + 2\sqrt{2}$
$a + b + c = 2 + 4 + 2= \boxed{8}$

Sep 5, 2021
edited by HighSchoolDx  Sep 5, 2021
#2
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Where is F in the Pentagon ABCDE?

Sep 5, 2021
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Good question.

Melody  Sep 6, 2021